Eccentricity of an ellipse

Average Error: 14.5 → 0.0

Time: 6.0s

Precision: binary64

Cost: 7040

\[\left(0 \leq b \land b \leq a\right) \land a \leq 1\]

Math

\[\sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|} \]

↓

\[{\left(1 - \frac{b}{a} \cdot \frac{b}{a}\right)}^{0.5} \]

FPCore

(FPCore (a b)
 :precision binary64
 (sqrt (fabs (/ (- (* a a) (* b b)) (* a a)))))

↓

(FPCore (a b) :precision binary64 (pow (- 1.0 (* (/ b a) (/ b a))) 0.5))

C

double code(double a, double b) {
	return sqrt(fabs((((a * a) - (b * b)) / (a * a))));
}

↓

double code(double a, double b) {
	return pow((1.0 - ((b / a) * (b / a))), 0.5);
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = sqrt(abs((((a * a) - (b * b)) / (a * a))))
end function

↓

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = (1.0d0 - ((b / a) * (b / a))) ** 0.5d0
end function

Java

public static double code(double a, double b) {
	return Math.sqrt(Math.abs((((a * a) - (b * b)) / (a * a))));
}

↓

public static double code(double a, double b) {
	return Math.pow((1.0 - ((b / a) * (b / a))), 0.5);
}

Python

def code(a, b):
	return math.sqrt(math.fabs((((a * a) - (b * b)) / (a * a))))

↓

def code(a, b):
	return math.pow((1.0 - ((b / a) * (b / a))), 0.5)

Julia

function code(a, b)
	return sqrt(abs(Float64(Float64(Float64(a * a) - Float64(b * b)) / Float64(a * a))))
end

↓

function code(a, b)
	return Float64(1.0 - Float64(Float64(b / a) * Float64(b / a))) ^ 0.5
end

MATLAB

function tmp = code(a, b)
	tmp = sqrt(abs((((a * a) - (b * b)) / (a * a))));
end

↓

function tmp = code(a, b)
	tmp = (1.0 - ((b / a) * (b / a))) ^ 0.5;
end

Wolfram

code[a_, b_] := N[Sqrt[N[Abs[N[(N[(N[(a * a), $MachinePrecision] - N[(b * b), $MachinePrecision]), $MachinePrecision] / N[(a * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]

↓

code[a_, b_] := N[Power[N[(1.0 - N[(N[(b / a), $MachinePrecision] * N[(b / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision]

Derivation

1. Initial program 14.5

   \[\sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|} \]

2. Applied egg-rr 0.0

   \[\leadsto \color{blue}{{\left(1 - {\left(\frac{b}{a}\right)}^{2}\right)}^{0.5}} \]

3. Taylor expanded in b around 0 14.5

   \[\leadsto {\color{blue}{\left(1 + -1 \cdot \frac{{b}^{2}}{{a}^{2}}\right)}}^{0.5} \]

4. Simplified 0.0

   \[\leadsto {\color{blue}{\left(1 - \frac{b}{a} \cdot \frac{b}{a}\right)}}^{0.5} \]
   Proof
   (-.f64 1 (*.f64 (/.f64 b a) (/.f64 b a))): 0 points increase in error, 0 points decrease in error
   (-.f64 1 (Rewrite<= times-frac_binary64 (/.f64 (*.f64 b b) (*.f64 a a)))): 52 points increase in error, 0 points decrease in error
   (-.f64 1 (/.f64 (Rewrite<= unpow2_binary64 (pow.f64 b 2)) (*.f64 a a))): 0 points increase in error, 0 points decrease in error
   (-.f64 1 (/.f64 (pow.f64 b 2) (Rewrite<= unpow2_binary64 (pow.f64 a 2)))): 0 points increase in error, 0 points decrease in error
   (Rewrite<= unsub-neg_binary64 (+.f64 1 (neg.f64 (/.f64 (pow.f64 b 2) (pow.f64 a 2))))): 0 points increase in error, 0 points decrease in error
   (+.f64 1 (Rewrite<= mul-1-neg_binary64 (*.f64 -1 (/.f64 (pow.f64 b 2) (pow.f64 a 2))))): 0 points increase in error, 0 points decrease in error

5. Final simplification 0.0

   \[\leadsto {\left(1 - \frac{b}{a} \cdot \frac{b}{a}\right)}^{0.5} \]

Alternatives

Alternative 1
Error	0.6
Cost	6976

\[\mathsf{fma}\left(\frac{-0.5}{a}, \frac{b}{\frac{a}{b}}, 1\right) \]

Alternative 2
Error	1.2
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2022297 
(FPCore (a b)
  :name "Eccentricity of an ellipse"
  :precision binary64
  :pre (and (and (<= 0.0 b) (<= b a)) (<= a 1.0))
  (sqrt (fabs (/ (- (* a a) (* b b)) (* a a)))))